The film drainage problem in droplet coalescence

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Abstract
A small drop of liquid 1 falls through a less dense liquid 2 and approaches the horizontal interface between liquid 2 and an underlying layer of liquid 1. After a short time the drop will be brought to rest (or nearly) in a hollow in the interface. Before the drop can coalesce with its bulk phase, the thin film of liquid 2 trapped between them must be squeezed out, and become sufficiently thin that rupture can occur. This is the film drainage problem. Early calculations, based on simple lubrication theory, fail to take proper account of two effects which are investigated here and shown to be decisive. They are the circulation induced in the drop and in the lower bulk fluid, which tends to speed up drainage, and the constriction in the film thickness at its periphery, which tends to slow it down. This constriction has been observed and some existing theories have attempted to model it in an ad hoc manner. We give here a physical explanation and calculate the minimum thickness explicitly. The effect of circulation in the adjacent fluids is also calculated.